MATH 251Examination I INovember 4, 2008Name:Student Number:Section:This exam has 13 questions for a total of 100 points. In order to obtain full credit for partial creditproblems, all work must be shown. Credit will not be given for an answer not supported bywork. The point value for each question is in parentheses to the right of the question number.You may not use a calculator on this exam.Please turn off and put away your cell phone.1:2:3:4:5:6:7:8:9:10:11:12:13:Total:Do not write in this box.MATH 251 EXAMINATION II November 4, 20081. (5 points) Match the sketches of phase portraits for 2x2 homogeneous linear systems x′= Ax with thenames of their critical points at the origin.1 23 45 6(a) saddle(b) node(c) proper node(d) center(e) spiralPage 2 of 9MATH 251 EXAMINATION II November 4, 20082. (5 points) Match the following formulas for general solutions of 2x2 homogeneous linear sys tems x′= Axwith the sketches of the phase portraits given in Problem 1:(a) c1e2t1−1+ c2e−t11(b) c1etcos t2 sin t+ c2etsin t2 cos t(c) c1e−t10+ c2e−t01(d) c1et10+ c2e−t01(e) c1cos t2 sin t+ c2sin t2 cos t3. (5 points) Match the three adj ectives for the critical point00of a 2 × 2 homogeneous linear systemsx′= Ax with the five general solutions given in the table below by placing one of the letters A, U, or S ineach of the five blanks.(a) c1e2t1−1+ c2e−t11Use(b) c1etcos t2 sin t+ c2etsin t2 cos tA for asymptotically stable(c) c1e−t10+ c2e−t01U for unstable(d) c1et10+ c2e−t01S for stable(e) c1cos t2 sin t+ c2sin t2 cos tPage 3 of 9MATH 251 EXAMINATION II November 4, 20084. (5 points) When an object with mass 5 kg is attached to a spring, the object stretches the spring by 2 m.A damper with damping coefficient of 4 N-s/m is attached to the system. Assume there is no external forceacting on the system and that acceleration du e to gravity is 10 m/s2. If the object is released 1 m aboveits equilibrium position and is given an initial downward velocity of 3 m/s. Assuming that the downwarddirection is positive, which initial value problem describes the motion of the system?(a) 5y′′+ 4y′+ 25y = 0, y(0) = 1, y′(0) = −3(b) 5y′′+ 4y′+ 25y = 0, y(0) = −1, y′(0) = 3(c) 5y′′+ 4y′+ 10y = 0, y(0) = 1, y′(0) = −3(d) 5y′′+ 4y′+ 10y = 0, y(0) = −1, y′(0) = 35. (5 points) Initially an object with mass 2 kg is located on a frictionless surface and is moving eastward withno forces acting on it. At time t = 3 seconds a constant external eastward force of .17 Newtons is added.At t = 4 seconds the object is struck with a hammer in such a fash ion that its momentum is reduced by.34 kg-m/s at that time. Which of the following F (t) represents the combination of these two forces.(a) F (t) = .17u(t − 3) + .34δ(t − 4)(b) F (t) = .17δ(t − 3) + .34u(t − 4)(c) F (t) = .17u(t − 3) − .34δ(t − 4)(d) F (t) = −.17δ(t − 3) + .34u(t − 4)6. (5 points) What is the general real-valued solution of the following ODE:y′′′+ y′= 0(a) y(t) = c1et+ c2e−t+ c2(b) y(t) = c1cos t + c2sin t + c2et(c) y(t) = c1+ c2cos t + c3sin t(d) y(t) = c1t2+ c2t + c3Page 4 of 9MATH 251 EXAMINATION II November 4, 20087. (5 points) A spr ing-mass system is described by th e differential equation y′′+ 2y′+ 10y = 0 with initialconditions y(0) = 1 and y′(0) = −1 Which s tatement will best describe the behavior of the solution y(t)after a long time?(a) y(t) will oscillate with increasing amplitude.(b) limt→∞y(t) = 0(c) y(t) will approach a periodic function with period2π3.(d) y(t) will approach a periodic function with period π.8. (5 points)R∞0e−(s−2)tdt is the Laplace transform of wh ich of the following functions?(a) t2(b) e2t(c) u(t − 2)(d) δ(t − 2)9. (5 points) If f (t) = u(t − 2) − 2u(t − 3) + 3u(t − 4) − 4u(t − 5), what is f (π)?(a) -2(b) -1(c) 0(d) 1Page 5 of 9MATH 251 EXAMINATION II November 4, 200810. A spring-mass system is modeled by the initial value problem2y′′+ γy′+ 18y = F (t), y(0) = 3, y′(0) = −12.(a) (6 points) If γ = 0 and F (t) = 0, what is the amplitude of displacement?(b) (3 points) If γ = 0 and F (t) = 2 cos(ωt), for which value(s) of ω will the system undergo resonance?(c) (3 points) If F (t) = 0, for which value(s) of γ will the system not oscillate?Page 6 of 9MATH 251 EXAMINATION II November 4, 200811. Compute the following:(a) (7 points)L{e3tu(t − 2)}.(b) (7 points)L−1ss2+ 8s + 25.Page 7 of 9MATH 251 EXAMINATION II November 4, 200812. (16 points) Use Laplace transforms to solve the following initial value prob lem:y′+ 3y = tu(t − 2) + δ(t − 3), y(0) = 1.Page 8 of 9MATH 251 EXAMINATION II November 4, 200813. Consider the 2 × 2 linear homogeneous sys tem:x′=0 4−1 0x.(a) (2 points) Find the vector x′at the point12.(b) (11 points) Find the solution x(t) that satisfies the initial condition x(0) =−10.Page 9 of
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